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Math Help - Rolle's Theorem Problem

  1. #1
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    Rolle's Theorem Problem

    Let f be a twice differentiable function on R with f continuous on [0, 1] such that
    the intergral of f(x) dx from (0, 1) = 2 x intergral of f(x) dx from (1/4, 3/4).
    Prove that there exists an xo in (0, 1) such that f(xo) = 0.


    Here is what I have so far....

    the intergral of f(x) dx from (0, 1) = 2 x intergral of f(x) dx from (1/4, 3/4)
    f(c1) (1-0) = 2 f(c2) (3/4 - 1/4)
    for c1 belonging to (0, 1) and c2 belonging to (1/4, 3/4)
    f(c1) = f(c2)
    By Rolle's Theorem, there exists c3 that belongs to (c1, c2) such that f ' (c3)=0

    I need another part similar to this so that the derivitive of these which would be f " (xo) = 0.

    Any suggestions are welcome.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Many people here will refuse to read non-LaTeX math.

    Quote Originally Posted by page929 View Post
    Let f be a twice differentiable function on \mathbb{R} with f'' continuous on [0,1] such that \int_0^1 f(x)\text{ }dx=2\int_{\frac{1}{4}}^{\frac{3}{4}}f(x)\text{ }dx. Prove that there exists some x_0\in(0,1) such that f''(x_0)=0
    Is this the question?
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  3. #3
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    Yes
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